Week 4 Study Notes on Scatterplots, Associations, and Correlation

Week 4 Overview

This week focuses on Scatterplots, Associations, and Correlation as explored in Chapter 6 of the textbook "Business Statistics" by Sharpe, Velleman, and Wright (Fourth Canadian Edition). The course is part of MATH 10008 - Statistics for Global Business Management.

Key Components for Relationship Analysis

To effectively analyze relationships between variables, we need to consider the following components:

1. Plot/Graph: Visualization tool to present the relationship between variables.

2. Characteristics to Describe: Identifying various attributes of the relationship to examine.

3. Measures of Characteristics: Quantifying the characteristics identified.

4. Inference Methods: Techniques to draw conclusions about the relationship.

   • Example Used: Investigating whether wine consumption results in a decrease in heart disease.

   • Response Variable: Death rate from heart disease (measures per 100,000 population).

   • Explanatory Variable: Wine consumption measured in liters per year per person.

   • This investigation highlights that we are dealing with bivariate analysis, a branch of statistics focusing on two variables.

Visualization and Description of Data

When examining scatterplots and data distributions, consider:

1. Form: Shapes like linear, curved, or cluster formations.

2. Direction: Connects to the trend line:

   • Positive: Indicates that as one variable increases, the other does as well.

   • Negative: Implies that as one variable increases, the other decreases.

3. Strength: Denotes how closely the points cling to the trend line.

4. Outliers: Points that deviate drastically from the pattern in the data.

   • Outliers can affect the strength assessment and must be analyzed critically.

Measuring the Strength of Relationships

To quantify the strength of a linear relationship between two quantitative variables, we utilize:

• Standardized (z) values: Transforming each variable into a standard score for analysis.

• Correlation Coefficient (r): This is defined as a measure of the direction (sign) and strength (magnitude) of a linear relationship between two quantitative variables.

  • It can take on values from $-1$ to $+1$, where:

    • A value of 1 indicates a perfect positive linear correlation.

    • A value of -1 indicates a perfect negative linear correlation.

    • A value of 0 indicates no linear correlation.

  • The formula for Pearson's correlation coefficient is given as:
    $r = rac{n( ext{Σ}(xy)) - ( ext{Σ}x)( ext{Σ}y)}{ ext{sqrt}[(n( ext{Σ}x^2) - ( ext{Σ}x)^2)(n( ext{Σ}y^2) - ( ext{Σ}y)^2)]}$

Example of Correlation Evaluation

• Example of correlation concerning wine consumption and heart disease:

  • A reported correlation value of $r = -0.843$ indicates a strong inverse relationship.

Scatterplots and their Purpose

Scatterplots are a primary tool for visualizing associations between quantitative variables. An example presented is the correlation between the monthly Canadian/U.S. exchange rate and oil prices, alongside a discussion on correlation versus causation.

• Key Note: Correlation does not imply causation. Just because two variables appear to be correlated (e.g., money wagered at US race tracks with a correlation $r = 0.931$) does not confirm that one variable causes the change in the other.

Lurking Variables

• Definition: Lurking variables are variables that have an effect on both studied variables but are not included in the analysis.

• Examples:

  1. A positive correlation between teachers' salaries and liquor sales does not imply that higher salaries cause increased liquor sales, indicating a need to explore other potential influences.

  2. Health data such as life expectancy correlates with the number of doctors per person in various countries with a correlation of $0.705$.

Correlation Tables

Correlation tables provide a compact summary of relationships across a set of variables. For instance, a table showing correlations among variable attributes for a sample of Amazon books, highlighting:

• The correlation coefficients from the table illustrate the relationships, where for example:

  • #Pages and Thick have a significant positive correlation of $0.813$.

Practical Application of Correlation Analysis

A series of analyses and quizzes following the correlation methodology report statistics:

• Example Analyses include:

  • Score and Time Spent: A correlation coefficient calculated at approximately $0.00596$ suggests a negligible relationship.

  • Time to Due Date and Score: Another weak correlation noted at $0.00498$.

Transforming Scatterplots to Assess Correlation

In practice, when scatterplots show non-linear associations, transformations are applied to make correlations more linear.

• Logarithmic Transformation: As shown in an example from Germany's solar power cost data from 2009-2013 demonstrating increased linear association after transformation.

• Caution: When interpreting transformed data, one must ensure clarity in representation to avoid misrepresentation of findings.

Homework and Practice

• Homework involves preparing a data report related to scatterplots, associations, and correlation analysis as outlined in Assignment 5, detailing tutorials and practical activities related to Chapter 6 on these topics.